To make the team, you are going to have to do 89 sit-ups for the coach a week from today. You decide to work up to it. You will start by doing 3 sit-ups today (no sense rushing into things) and end on the 8th day with 89. You don’t know how many you will do tomorrow, but you decide that from the 3rd day on, the number of sit-ups you do will be the sum of what you did on the two preceding days. That is, the number you do on Wednesday will be the sum of the number you did on Monday and the number you did on Tuesday; the number you do on Thursday will be the sum of what you did on Tuesday and Wednesday, and so on. Find out how many sit-ups you should do tomorrow to make this work, so that you come out with 89 a week from today.

## Identifying Patterns

## Six Sequences to Continue

Continue these sequences. Explain how each one works.

a. | 0 | 1 | 3 | 6 | 7 | 9 | 12 | 13 | ___ | ___ | ___ |

b. | 1 | 2 | 3 | 4 | 9 | 8 | 27 | 16 | ___ | ___ | ___ |

c. | 0 | 4 | 8 | 4 | 8 | 12 | 6 | 10 | ___ | ___ | ___ |

d. | 1 | 2 | 3 | 3 | 6 | 9 | 9 | ___ | ___ | ___ | ___ |

e. | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | ___ | ___ | ___ |

f. | 0 | 1 | 1 | 2 | 4 | 7 | 13 | 24 | ___ | ___ | ___ |

## Unusual Sequence

Here is a rather unusual sequence: 125, 59, 42, 34, 14, 4, ____. Can you finish the sequence?

## One Sequence to Continue

What is the next term in this sequence: 2, 9, 28, 65, 126, 217, _____?

## Alternating Sums of Squares

Evaluate 1^{2} – 2^{2} + 3^{2} – 4^{2} + 5^{2} – 6^{2} + … + 199^{2}.

## Giant Binomial Squared

The sum of the digits in base ten of (10^{(4n}^{2} +8) +1)^{2}, where *n* is a positive integer, is:

- 4
- 4
*n* - 2 + 2
*n* - 4
*n*^{2} *n*^{2}+*n*+ 2

## Defined Operations 2; Do Composition

If 3 *h* = 10, 7 *h* = 50, 5 *h* = 26; and 4 *b* = 1, 7 *b* = 2.5, 20 *b* = 9, then what is *n*

if *n hb* = 17.5?

## Defined Operations 3; Do Composition

If 10 *g* = 16, 100 *g* = 196, 4 *g* = 4 and 12 *r* = 5; 300 *r* = 101, 30 *r* = 11, then what is *n* if *n* *g* *r* = 9?

## Function as Average

Let *x*_{k} = (-1)^{k} for any positive integer *k*. Let *f*(*n*) = (*x*_{1} + *x*_{2} + … + *x*_{n})/*n*, where *n* is a positive integer. Give the range of this function.

- 0
- 1/
*n*(where*n*is any positive integer) - 0 and -1/
*n*(where*n*is any odd positive integer) - 0 and 1/
*n*(where*n*is any positive integer) - 1 and 1/
*n*(where*n*is any odd positive integer).

## Big f(x), Find f(6)

If *f*(*n*) is a function such that *f*(1) = *f*(2) = *f*(3) = 1, and such that:

*n*> 3, then

*f*(6) is equal to:

- 2
- 3
- 7
- 11
- 26.